Rewriting a symmetric cubic inequality

Proved

WorkbookSource.problem_43229

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

For real $a,b,c$ , set $S=ab(a+b)+bc(b+c)+ca(c+a)$ . Then

S\ge\frac34(a+b)(b+c)(c+a)\quad\Longleftrightarrow\quad S\ge6abc.

Source: InternLM Lean-Workbook, record lean_workbook_43229 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_43229  (a b c : ℝ) :
  a * b * (a + b) + b * c * (b + c) + c * a * (c + a) ≥
  (3 * (a + b) * (b + c) * (c + a)) / 4 ↔
  a * b * (a + b) + b * c * (b + c) + c * a * (c + a) ≥
  6 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_43229; Apache-2.0

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